Existence Of Multiple Positive Solutions For Nonlinear Elliptic Equations With Critical Exponent

In this paper, the following nonlinear elliptic equation withcritical exponent involving the p-Laplacian is discussed,where q = (N_p/N-p) -1(N > p≥2) is the critical Sobolev's exponent,μ≥0 isa given constant,φ(x)∈L1(RN)∩Cα(RN),Δp is the p-Laplacian operator.The existence of multiple positive solutions for above problem has been studied,without the increase restriction on the nonlinear function f.(On normal case,such studies are to be based on the assumption of increase restriction on thenonlinear function f.) Firstly, on the basis of the supersolution and subsolution,by using the maximun principle, iteration method and standard barrier method,the following results are proved: there exists a constantμ~* such that aboveequation possesses a minimal positive solution ifμ<μ~*; equation possessesa unique solution ifμ=μ~*; there are no solution ifμ>μ~*. At the sametime, the range of the value forμ~* has been confirmed. Furthermore, accordingto the minimal positive solution, by using the Mountain Pass Lemma and theconcentration-compactness principle, the existence of second positive solution issubsequently proved.